
More concretely, the \(\mathinner {|{+}\rangle }_{\rm {gkp}}\)  Wigner function has rows of positive peaks periodically arranged in phase space along even integer multiples of \(\sqrt{\pi }\)  in the \(p\)  quadrature, and alternating positive and negative peaks for odd multiples (see Fig. 1a of [1]}).
The broad distribution in \(q\)  from \(\mathbf {Y}_{\rm p}\)  causes the rows of positive and negative peaks to cancel, and the rows of positive-only peaks to add, washing away the Wigner negativity and yielding a distribution mimicking a mixture of \(p\)  squeezed states spaced by even multiples of \(\sqrt{\pi }\)  in \(p\) .
While this is not a true \(p\) -squeezed state, we do not expect it to provide an underestimation of the error probability of the quantum memory, especially since a mixture of states (as opposed to a pure \(p\) -squeezed state) would only add more noise and hence make the decoding problem more difficult.
